pgr6:=function()
local z,g,sg,p;
z:=FreeGroup("a","b");
g:=z/[z.1^3,z.2^2,z.1*z.2*z.1*z.2*z.1*z.2*z.1^2*z.2*z.1^2*z.2*z.1^2*z.2];
sg:=Subgroup(g,[g.1]);
p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
return p;
end;
G6:=pgr6(); gap> Size(G6); 48 gap> N6:=NormalSubgroups(G6);; gap> NTSize(N6); Groesse des 1. Normalteilers: 1 Groesse des 2. Normalteilers: 2 Groesse des 3. Normalteilers: 4 Groesse des 4. Normalteilers: 8 Groesse des 5. Normalteilers: 16 Groesse des 6. Normalteilers: 24 Groesse des 7. Normalteilers: 48 gap> Centre(G6); Group([ ( 1,14,16,10)( 2,11,15, 7)( 3, 6,12, 9)( 4, 8,13, 5) ]) gap> Size(Centre(G6)); 4 gap> IsSolvable(G6); true gap> CompositionSeries(G6); [ Group([ ( 1, 2)( 3, 5)( 4, 6)( 7,10)( 8,12)( 9,13)(11,14)(15,16), ( 2, 3, 4)( 5, 7, 9)( 6, 8,11)(12,13,15), ( 1, 8,16, 5)( 2, 3,15,12)( 4,14,13,10)( 6, 7, 9,11), ( 1, 6,16, 9)( 2,13,15, 4)( 3,14,12,10)( 5, 7, 8,11), ( 1,16)( 2,15)( 3,12)( 4,13)( 5, 8)( 6, 9)( 7,11)(10,14) ]), Group([ ( 2, 3, 4)( 5, 7, 9)( 6, 8,11)(12,13,15), ( 1, 8,16, 5)( 2, 3,15,12)( 4,14,13,10)( 6, 7, 9,11), ( 1, 6,16, 9)( 2,13,15, 4)( 3,14,12,10)( 5, 7, 8,11), ( 1,16)( 2,15)( 3,12)( 4,13)( 5, 8)( 6, 9)( 7,11)(10,14) ]), Group([ ( 1, 8,16, 5)( 2, 3,15,12)( 4,14,13,10)( 6, 7, 9,11), ( 1, 6,16, 9)( 2,13,15, 4)( 3,14,12,10)( 5, 7, 8,11), ( 1,16)( 2,15)( 3,12)( 4,13)( 5, 8)( 6, 9)( 7,11)(10,14) ]), Group([ ( 1, 6,16, 9)( 2,13,15, 4)( 3,14,12,10)( 5, 7, 8,11), ( 1,16)( 2,15)( 3,12)( 4,13)( 5, 8)( 6, 9)( 7,11)(10,14) ]), Group([ ( 1,16)( 2,15)( 3,12)( 4,13)( 5, 8)( 6, 9)( 7,11)(10,14) ]), Group(()) ]
gap> ct6:=CharacterTable(G6); CharacterTable( Group([ ( 2, 3, 4)( 5, 7, 9)( 6, 8,11)(12,13,15), ( 1, 2)( 3, 5)( 4, 6)( 7,10)( 8,12)( 9,13)(11,14)(15,16) ]) ) gap> Display(ct6); CT1 2 4 2 2 3 2 2 2 3 2 4 2 2 4 4 3 1 1 1 . 1 1 1 . 1 1 1 1 1 1 1a 3a 3b 2a 12a 12b 6a 4a 6b 4b 12c 12d 4c 2b 2P 1a 3b 3a 1a 6a 6b 3a 2b 3b 2b 6a 6b 2b 1a 3P 1a 1a 1a 2a 4b 4c 2b 4a 2b 4c 4c 4b 4b 2b 5P 1a 3b 3a 2a 12d 12c 6b 4a 6a 4b 12b 12a 4c 2b 7P 1a 3a 3b 2a 12c 12d 6a 4a 6b 4c 12a 12b 4b 2b 11P 1a 3b 3a 2a 12b 12a 6b 4a 6a 4c 12d 12c 4b 2b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 1 X.3 1 A /A 1 A /A /A 1 A 1 A /A 1 1 X.4 1 A /A -1 -A -/A /A 1 A -1 -A -/A -1 1 X.5 1 /A A 1 /A A A 1 /A 1 /A A 1 1 X.6 1 /A A -1 -/A -A A 1 /A -1 -/A -A -1 1 X.7 2 -/A -A . B /B A . /A D -B -/B -D -2 X.8 2 -/A -A . -B -/B A . /A -D B /B D -2 X.9 2 -1 -1 . C -C 1 . 1 D -C C -D -2 X.10 2 -1 -1 . -C C 1 . 1 -D C -C D -2 X.11 2 -A -/A . -/B -B /A . A D /B B -D -2 X.12 2 -A -/A . /B B /A . A -D -/B -B D -2 X.13 3 . . -1 . . . -1 . 3 . . 3 3 X.14 3 . . 1 . . . -1 . -3 . . -3 3 A = E(3) = (-1+ER(-3))/2 = b3 B = -E(12)^11 C = -E(4) = -ER(-1) = -i D = -2*E(4) = -2*ER(-1) = -2i gap> quit;